A371711 - OEIS

%I #22 Apr 08 2024 18:50:30

%S 1,2,2,0,2,0,-4,0,22,0,-36,0,-108,0,376,0,1894,0,-4180,0,-10628,0,

%T 25032,0,-300676,0,709528,0,2579816,0,-9878288,0,85643910,0,

%U -153029428,0,-514327828,0,2085333416,0,-10936571628,0,17723495752,0,60604096152,0,-268963402096,0,-1956018724132,0,5018953346552

%N Expansion of g.f. A(x) satisfying A(x) = 2*x + A(2*x^2)^(1/2) with A(0) = 1.

%H Paul D. Hanna, <a href="/A371711/b371711.txt">Table of n, a(n) for n = 0..518</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(2*x^2) = (A(x) - 2*x)^2.

%F (2) A(8*x^4) = (A(2*x^2) - 4*x^2)^2 = ((A(x) - 2*x)^2 - 4*x^2)^2.

%F (3) A(128*x^8) = (A(8*x^4) - 16*x^4)^2 = (((A(x) - 2*x)^2 - 4*x^2)^2 - 16*x^4)^2.

%F (4) 1 = (...(((((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 - (2*x)^16)^2 - (2*x)^32)^2 - (2*x)^64)^2 - ...), an infinite nested square.

%F (5) A(x) = lim_{n->oo} (2*x + sqrt((2*x)^2 + sqrt((2*x)^4 + sqrt((2*x)^8 + sqrt((2*x)^16 + sqrt((2*x)^32 + sqrt((2*x)^64 + ... + sqrt((2*x)^(2^n) + 1)...))))))), with 1 appearing in the final (n-th) nested radical only.

%F (6) A(x) = x/Series_Reversion(F(x)), where F(x) = x*A(F(x)) is the g.f. of A371712 defined by F( 2*x^2/(1-2*x)^2 ) = 2*F(x)^2.

%F (7) F( 2*x^2/(A(x) - 2*x)^2 ) = 2*x^2, where F(x/A(x)) = x and F(x) = x*A(F(x)) is the g.f. of A371712.

%F The radius of convergence is r = 1/2, with A(r) = (3 + sqrt(5))/2 = 2.6180339...

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^4 - 4*x^6 + 22*x^8 - 36*x^10 - 108*x^12 + 376*x^14 + 1894*x^16 - 4180*x^18 - 10628*x^20 + ...

%e where A(2*x^2) = (A(x) - 2*x)^2.

%e The g.f. A(x) satisfies the infinite nested square given by

%e 1 = (...(((((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 - (2*x)^16)^2 - (2*x)^32)^2 - (2*x)^64)^2 - ...)

%e as illustrated in the following process.

%e (1) (A(x) - 2*x)^2 = 1 + 4*x^2 + 8*x^4 + 32*x^8 - 256*x^12 + 5632*x^16 - 36864*x^20 - 442368*x^24 + ...

%e (2) ((A(x) - 2*x)^2 - (2*x)^2)^2 = 1 + 16*x^4 + 128*x^8 + 8192*x^16 - 1048576*x^24 + 369098752*x^32 + ...

%e (3) (((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 = 1 + 256*x^8 + 32768*x^16 + 536870912*x^32 - 17592186044416*x^48 + ...

%e (4) ((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 = 1 + 65536*x^16 + 2147483648*x^32 + 2305843009213693952*x^64 + ...

%e One may continue indefinitely in this manner, the process tending to 1 as a limit for |x| <= 1/2.

%e RELATED SERIES.

%e Let F(x) be the g.f. of A371712, then F(x/A(x)) = x, where

%e F(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 74*x^5 + 292*x^6 + 1204*x^7 + 5112*x^8 + 22182*x^9 + 97964*x^10 + 439252*x^11 + 1995864*x^12 + ...

%e which satisfies F( 2*x^2/(1-2*x)^2 ) = 2*F(x)^2.

%o (PARI) /* By definition, A(x) = 2*x + A(2*x^2)^(1/2) with A(0) = 1 */

%o {a(n) = my(A=1); for(i=1,#binary(n), A = 2*x + sqrt(subst(A,x,2*x^2) +O(x^(n+2))) ); polcoeff(A, n)}

%o for(n=0, 50, print1(a(n), ", "))

%Y Cf. A371712.

%K sign

%O 0,2

%A _Paul D. Hanna_, Apr 07 2024